4a^2-1=7a

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Solution for 4a^2-1=7a equation:


Simplifying
4a2 + -1 = 7a

Reorder the terms:
-1 + 4a2 = 7a

Solving
-1 + 4a2 = 7a

Solving for variable 'a'.

Reorder the terms:
-1 + -7a + 4a2 = 7a + -7a

Combine like terms: 7a + -7a = 0
-1 + -7a + 4a2 = 0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-0.25 + -1.75a + a2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -1.75a + 0.25 + a2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -1.75a + a2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -1.75a + a2 = 0 + 0.25
-1.75a + a2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-1.75a + a2 = 0.25

The a term is -1.75a.  Take half its coefficient (-0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
-1.75a + 0.765625 + a2 = 0.25 + 0.765625

Reorder the terms:
0.765625 + -1.75a + a2 = 0.25 + 0.765625

Combine like terms: 0.25 + 0.765625 = 1.015625
0.765625 + -1.75a + a2 = 1.015625

Factor a perfect square on the left side:
(a + -0.875)(a + -0.875) = 1.015625

Calculate the square root of the right side: 1.007782219

Break this problem into two subproblems by setting 
(a + -0.875) equal to 1.007782219 and -1.007782219.

Subproblem 1

a + -0.875 = 1.007782219 Simplifying a + -0.875 = 1.007782219 Reorder the terms: -0.875 + a = 1.007782219 Solving -0.875 + a = 1.007782219 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + a = 1.007782219 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + a = 1.007782219 + 0.875 a = 1.007782219 + 0.875 Combine like terms: 1.007782219 + 0.875 = 1.882782219 a = 1.882782219 Simplifying a = 1.882782219

Subproblem 2

a + -0.875 = -1.007782219 Simplifying a + -0.875 = -1.007782219 Reorder the terms: -0.875 + a = -1.007782219 Solving -0.875 + a = -1.007782219 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + a = -1.007782219 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + a = -1.007782219 + 0.875 a = -1.007782219 + 0.875 Combine like terms: -1.007782219 + 0.875 = -0.132782219 a = -0.132782219 Simplifying a = -0.132782219

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.882782219, -0.132782219}

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